It should be emphasized that the canonical transformations as
described in terms of the position and momentum operators
[1]}, [2]}, [3]} are also special cases of real/complex number
canonical transformations of the (projective) Hilbert space as
a symplectic/Kähler manifold [4]}. They are special in
the sense that they are taking the noncommutative canonical
coordinates to noncommutative canonical coordinates with
truly nontrivial changes in the noncommutative values. Of
course here we are talking about the phase space of the
composite system. Note
that a noncommutative value is really an element of the
noncommutative algebra of such values, and as such is a fixed
abstract quantity as kind of a noncommutative analog of a real
number. However, any particular representation of it as a
sequence of complex numbers, we can see, is dependent on
the choice of basis of the Hilbert space. Especially in relation
to the quantum reference frame transformations discussed here,
we have issues about choice of coordinates at different levels.
The transformations can be seen as transformations of the set
of noncommutative coordinates, hence changes of their
noncommutative values. The set of noncommutative coordinates
is also a set of basic observables to which any observable in the
observable algebra is like a function of. The transformations
hence change the representation of all observables, and their
noncommutative values. The basis of the Hilbert space defined
using a set of eigenstates then changes along with it. To
compare two noncommutative values, we have to use sequence
representations based on the exact same basis, as done above.
